Although the herringbone pattern “1234”
is well known both at the nano- and microscales, the other patterns
shown in Fig. REF (b-d) have relatively rarely been observed
and reported [1]}, [2]}. A recent work by Tang et
al. [3]} reports a periodic
array of flux-closure domains in PbTiO\(_{3}\) . However, this polarization
pattern differs from the checkerboard pattern in Fig.Â REF d,
in that the observed pattern possesses a \(180^{\circ }\)  domain wall
at the vortex core. This domain pattern observed by Tang et
al. is referred in section III-C of this
paper, where the phase-field simulations indicate its stability in
the presence of external loads. 
